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Mirrors > Home > MPE Home > Th. List > prod0 | Structured version Visualization version GIF version |
Description: A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.) |
Ref | Expression |
---|---|
prod0 | ⊢ ∏𝑘 ∈ ∅ 𝐴 = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 11866 | . 2 ⊢ 1 ∈ ℤ | |
2 | nnuz 12134 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | id 22 | . . 3 ⊢ (1 ∈ ℤ → 1 ∈ ℤ) | |
4 | ax-1ne0 10459 | . . . 4 ⊢ 1 ≠ 0 | |
5 | 4 | a1i 11 | . . 3 ⊢ (1 ∈ ℤ → 1 ≠ 0) |
6 | 2 | prodfclim1 15086 | . . 3 ⊢ (1 ∈ ℤ → seq1( · , (ℕ × {1})) ⇝ 1) |
7 | 0ss 4276 | . . . 4 ⊢ ∅ ⊆ ℕ | |
8 | 7 | a1i 11 | . . 3 ⊢ (1 ∈ ℤ → ∅ ⊆ ℕ) |
9 | fvconst2g 6838 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = 1) | |
10 | noel 4222 | . . . . 5 ⊢ ¬ 𝑘 ∈ ∅ | |
11 | 10 | iffalsei 4397 | . . . 4 ⊢ if(𝑘 ∈ ∅, 𝐴, 1) = 1 |
12 | 9, 11 | syl6eqr 2851 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 1)) |
13 | 10 | pm2.21i 119 | . . . 4 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
14 | 13 | adantl 482 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
15 | 2, 3, 5, 6, 8, 12, 14 | zprodn0 15130 | . 2 ⊢ (1 ∈ ℤ → ∏𝑘 ∈ ∅ 𝐴 = 1) |
16 | 1, 15 | ax-mp 5 | 1 ⊢ ∏𝑘 ∈ ∅ 𝐴 = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ⊆ wss 3865 ∅c0 4217 ifcif 4387 {csn 4478 × cxp 5448 ‘cfv 6232 ℂcc 10388 0cc0 10390 1c1 10391 ℕcn 11492 ℤcz 11835 ∏cprod 15096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-prod 15097 |
This theorem is referenced by: prod1 15135 fprodf1o 15137 fprodcllem 15142 fprodmul 15151 fproddiv 15152 fprodfac 15164 fprodconst 15169 fprodn0 15170 fprod2d 15172 fprodmodd 15188 risefac0 15218 coprmprod 15838 coprmproddvds 15840 prmo0 16205 gausslemma2dlem4 25631 breprexp 31517 bcprod 32580 fprodexp 41438 fprodabs2 41439 mccl 41442 fprodcn 41444 fprodcncf 41747 dvmptfprod 41793 dvnprodlem3 41796 |
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